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100+ Free A-Level Mathematics Practice Questions

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The data 2, 4, 5, 7, 9 has mean 5.4. Find the variance.

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Key Facts: A-Level Mathematics Exam

A*-E

Grading scale

Ofqual

May-June

Exam series

AQA, Edexcel, OCR timetable

3 boards

Specifications available

AQA, Edexcel, OCR

100

Free practice questions here

OpenExamPrep

AQA, Edexcel, OCR A-Level Mathematics is assessed through linear end-of-course exam papers (Year 13). Coverage spans pure mathematics, statistics, mechanics, and grading uses the A*-E scale on 2026 specifications.

Sample A-Level Mathematics Practice Questions

Try these sample questions to test your A-Level Mathematics exam readiness. Each question includes a detailed explanation. Start the interactive quiz above for the full 100+ question experience with AI tutoring.

1Simplify (2x^3)^4 / (4x^5).
A.4x^7
B.8x^7
C.4x^17
D.2x^7
Explanation: (2x^3)^4 = 2^4 x^12 = 16x^12. Divide by 4x^5: 16x^12 / 4x^5 = 4x^(12-5) = 4x^7.
2Rationalise the denominator: 6 / (3 - sqrt(3)).
A.3 + sqrt(3)
B.(3 + sqrt(3))/2
C.3 - sqrt(3)
D.2(3 + sqrt(3))
Explanation: Multiply top and bottom by (3 + sqrt(3)): 6(3 + sqrt(3)) / (9 - 3) = 6(3 + sqrt(3))/6 = 3 + sqrt(3).
3The quadratic x^2 + (k+1)x + 4 = 0 has equal roots. Find the positive value of k.
A.3
B.5
C.1
D.-5
Explanation: Equal roots require discriminant = 0: (k+1)^2 - 16 = 0, so (k+1)^2 = 16, k+1 = +/-4. Positive value: k = 3.
4Write x^2 - 6x + 11 in the form (x - a)^2 + b.
A.(x - 3)^2 + 2
B.(x - 3)^2 + 11
C.(x - 6)^2 + 11
D.(x + 3)^2 + 2
Explanation: Complete the square: x^2 - 6x = (x - 3)^2 - 9. So x^2 - 6x + 11 = (x - 3)^2 - 9 + 11 = (x - 3)^2 + 2.
5Solve the simultaneous equations y = x + 3 and y = x^2 + 1. Find the positive x-coordinate of intersection.
A.2
B.1
C.3
D.-1
Explanation: Substitute: x + 3 = x^2 + 1, so x^2 - x - 2 = 0, (x - 2)(x + 1) = 0. Positive root: x = 2.
6Solve the inequality x^2 - x - 6 < 0.
A.-2 < x < 3
B.x < -2 or x > 3
C.-3 < x < 2
D.x > 3
Explanation: Factorise: (x - 3)(x + 2) < 0. The parabola opens upwards, so the expression is negative between the roots: -2 < x < 3.
7Find the range of values of k for which x^2 + kx + 9 = 0 has no real roots.
A.-6 < k < 6
B.k < -6 or k > 6
C.k > 6
D.0 < k < 9
Explanation: No real roots requires discriminant < 0: k^2 - 36 < 0, so k^2 < 36, giving -6 < k < 6.
8Simplify sqrt(50) + sqrt(18) - sqrt(8).
A.6 sqrt(2)
B.4 sqrt(2)
C.8 sqrt(2)
D.60
Explanation: sqrt(50) = 5 sqrt(2), sqrt(18) = 3 sqrt(2), sqrt(8) = 2 sqrt(2). Sum: 5 + 3 - 2 = 6, giving 6 sqrt(2).
9Evaluate 27^(-2/3).
A.1/9
B.1/27
C.9
D.-9
Explanation: 27^(1/3) = 3, so 27^(2/3) = 9. The negative exponent gives the reciprocal: 27^(-2/3) = 1/9.
10The quadratic f(x) = 2x^2 - 8x + 5 is written as 2(x - p)^2 + q. Find q.
A.-3
B.3
C.-1
D.5
Explanation: Factor 2: 2(x^2 - 4x) + 5 = 2[(x - 2)^2 - 4] + 5 = 2(x - 2)^2 - 8 + 5 = 2(x - 2)^2 - 3. So q = -3.

About the A-Level Mathematics Exam

A-Level Mathematics is offered by AQA, Edexcel, OCR as part of the UK A-Level qualification framework. The course covers pure mathematics, statistics, mechanics and is assessed primarily through written exam papers at the end of the two-year course.

Questions

100 scored questions

Time Limit

5-7 hours total across multiple papers

Passing Score

Grade E is the minimum pass, Grades A*-E count as a pass (A*-A-B-C-D-E)

Exam Fee

£75-£130 per subject (school-set entry fee) (AQA, Edexcel, OCR)

A-Level Mathematics Exam Content Outline

Core

Pure: Algebra and Functions

Polynomials, indices, surds, transformations, exponentials and logarithms

Core

Pure: Coordinate Geometry and Sequences

Lines, circles, parametric equations, binomial expansion, arithmetic and geometric sequences

Core

Pure: Trigonometry and Calculus

Trigonometric identities, radians, differentiation, integration, differential equations, numerical methods

Core

Pure: Vectors and Proof

2D and 3D vectors, dot product, proof techniques (deduction, contradiction, induction)

Core

Statistics

Sampling, data representation, probability, statistical distributions (binomial, normal), hypothesis testing

Core

Mechanics

Kinematics (suvat), forces and Newton's laws, moments, projectile motion, friction

How to Pass the A-Level Mathematics Exam

What You Need to Know

  • Passing score: Grade E is the minimum pass, Grades A*-E count as a pass (A*-A-B-C-D-E)
  • Exam length: 100 questions
  • Time limit: 5-7 hours total across multiple papers
  • Exam fee: £75-£130 per subject (school-set entry fee)

Keys to Passing

  • Complete 500+ practice questions
  • Score 80%+ consistently before scheduling
  • Focus on highest-weighted sections
  • Use our AI tutor for tough concepts

A-Level Mathematics Study Tips from Top Performers

1Use past papers from your specific exam board — questions follow the same style year on year
2Time yourself on full papers to build pacing for the long extended-response questions
3Build a clear understanding of mark schemes — examiners reward specific assessment objectives
4Review examiner reports each summer; common errors repeat

Frequently Asked Questions

What exam boards offer A-Level Mathematics?

A-Level Mathematics is offered by AQA, Edexcel, OCR. All boards follow Ofqual subject content but vary in the choice of set texts, optional topics, and paper structure.

When is the A-Level Mathematics exam taken?

Exams are written in the May-June series at the end of the two-year linear A-Level course. Most students sit the papers in Year 13.

How is A-Level Mathematics graded?

A-Levels are graded A*-E. A* is the highest grade and E is the minimum pass. UCAS tariff points are awarded for A-Level grades on most university applications.

How many papers does A-Level Mathematics have?

Most A-Level subjects have 3 written papers. The exact number, timing, and weighting depend on the chosen exam board. Some subjects also include a non-examined assessment (NEA) coursework component.